多尺度收缩与Lévy过程
Multiscale Shrinkage and Lévy Processes
摘要：
一个新的收缩为基础的建设开发的可压缩矢量$\boldsymbol{x}\in\mathbb{R}^n$，在这种情况下，$\xv$的组件自然与树结构。重要的示例是当$\xv$对应于数据的小波或块DCT表示的系数时。我们详细考虑的方法，并给出了数值结果，是基于增量的伽马过程。然而，我们证明，一般的框架是适当的许多其他类型的收缩先验，所有在Lévy过程的家庭，伽玛过程的特殊情况。贝叶斯推断是通过用MCMC算法的样本近似后验，以及通过构造后验的启发式变分近似来进行的。我们还考虑期望最大化（EM）的MAP（点）的解决方案。最先进的结果表现为压缩感知和去噪应用，后者具有尖峰（非高斯）噪声。
英文摘要：
A new shrinkage-based construction is developed for a compressible vector $\boldsymbol{x}\in\mathbb{R}^n$, for cases in which the components of $\xv$ are naturally associated with a tree structure. Important examples are when $\xv$ corresponds to the coefficients of a wavelet or block-DCT representation of data. The method we consider in detail, and for which numerical results are presented, is based on increments of a gamma process. However, we demonstrate that the general framework is appropriate for many other types of shrinkage priors, all within the Lévy process family, with the gamma process a special case. Bayesian inference is carried out by approximating the posterior with samples from an MCMC algorithm, as well as by constructing a heuristic variational approximation to the posterior. We also consider expectation-maximization (EM) for a MAP (point) solution. State-of-the-art results are manifested for compressive sensing and denoising applications, the latter with spiky (non-Gaussian) noise.
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